Any downtown hotel would be great. In my opinion rates are pretty reasonable compared to other big city downtown hotels, and worth being close (within walking distance or very near) to all the attractions. Alot of the downtown hotels also offer optional package deals that provide great discounts to the Rock Hall, Playhouse Square (second largest theater district outside of New York city), and other attractions. Call the hotels directly for this information. All the downtown hotels are pretty much within walking distance of both these attractions, so when looking at hotels just look on a map for its location. There are a few hotels east of E.12th street, and I would recommend avoiding those. Anything downtown between E.12th and W.9th is fine.
Perhaps the best recommendation in terms of location (especially for walking to the Rock Hall and Jacobs Field), quality, and value would be the Hyatt Hotel. This hotel makes use of one of Cleveland's most famous, historic, and beautiful buildings; The Old Arcade (be sure to see it anyway even if you don't stay there, and Vivo's and V Lounge in the Old Arcade is great restaurant and lounge; really cool).
If the Hyatt is still a little more than you planned on, the best budget hotel in my opinion is the Holiday Inn Express. This is in the same good downtown location as the Hyatt, and makes use of a historic bank building (beautiful lobby).
Of course the nicest hotels in Cleveland are the Ritz-Carlton and Renaissance Hotel. Both are connected to Tower City. Tower City is the downtown's transit hub, shopping center, and signature skyscaper located on downtown's focal point; Public Square. There are also a good food court and restaurants located here. These hotels are in a good location too, and easily within walking distance to Jacobs Field (via indoor walkway with 'Gateway' signage), and the Rock Hall. These hotels are a little pricey, but compared to other Ritz-Carlton and Renaissance Hotels they are quite the deal. If you have never stayed at these luxury hotel groups, than Cleveland is certainly the place to do it, because the rates for these brands/quality/service are certainly attractive.
Also, when staying downtown you should keep in mind that the Warehouse District (along W.6th, St. Clair, and W.9th) is the most nice, trendy, and most exciting dining and nightlife district downtown. There are a number of pretty classy and popular restaurants, bars, and clubs here. A lot of these places are more upscale, vallet parking, and have sidewalk patios. It is the premier place to "be seen" in Cleveland. I would recommend visiting. Another area downtown to visit is around the intersection of E.4th and Euclid. Vivo's, House of Blues (a great music venue and restaurant), and Pickwick and Frolic (a classy comedy, cabaret, and restaurant complex), Flannery's Irish pub, and soon Lola's (one of Cleveland's most nationally acclaimed restaurants) is located here.
If you are staying downtown and visiting the Rock Hall and Jacobs Field, you should also visit Tower City, Old Arcade, and the Civic Center (where you can see the beautiful public library, federal reserve bank, city/county administrations buildings, and convention center all along a grassy park-like mall). Outside of downtown (but very near) I highly recommend the West Side Market (a municipally owned and old-world European style market in Ohio City). In addition to the West Market Ohio City along W.25th offers great coffee shops, restaurants, bars, and the Great Lakes Brewing Company (one of the best breweries in the country)!!! Dortmunder Gold is my favorite brew. Check it out. Go to the bar, maybe the restaurant, and perhaps a tour of the brewery. It is great.
When visiting Cleveland also consider heading to University Circle. This district just east of downtown is Cleveland's second downtown. It is where all of the cultural, medical, and educational institutions are located and it is pretty serene with MANY museums and the country's highest rated orchestra at Severance Hall (gorgeous building). These insitutions are surrounded by a picturesque lagoon and park, and actually make up the highest concentration of cultural/medical/educational institutions within a square mile in the world. University Circle also borders Cleveland's Little Italy (around the intersection of Mayfield Rd. and Murray Hill Rd.) which is a great place with a lot of character for dining and browsing shops and art galleries.
If you have more time to kill in Cleveland I recommend heading to Shaker Square (beautiful European-style square with excellent light rail service to Tower City downtown and a great place for dinner and movie, and some unique shopping), or Coventry Village in Cleveland Heights (Cleveland's Greenwhich Village). Depending on where you are you are coming from, if you have a car, and if you would like to do so good shopping, I recommend the impressive shopping centers of Crocker Park, Beachwood Place, Legacy Village, and Eton Collection located not too far from outside the city center. They all have good websites and list all their shops and restaurants.
Enjoy your trip! And post additional comments if you'd like more information and suggestions.