One name can refer to both a town and a county.
Neunkirchen is the town that gives the county its name. The county also contains Ottweiler, Eppelborn, Illingen, Merchweiler, Schiffweiler and Spiesen-Elversberg. A town is only one part of the larger local territorial unit.
A second distinction helps: the county administration is based in Ottweiler. That does not mean you should choose Ottweiler when asked for the county’s name. The correct selection remains Neunkirchen.
The other three answers
The remaining county names belong to other federal states.
Mecklenburgische Seenplatte
Mecklenburgische Seenplatte is a county in Mecklenburg-Vorpommern. Here, the name does not describe a lake district in Saarland.
Altötting
Altötting is a county in Bavaria. A familiar county name does not by itself mean that it belongs to Saarland.
Rhein-Sieg-Kreis
Rhein-Sieg-Kreis belongs to Nordrhein-Westfalen. It is a different county from Neunkirchen in Saarland.
The question asks about the county level.
The town of Neunkirchen is not the state capital. The capital is Saarbrücken. County, town and federal state perform different functions and are not interchangeable terms.
Similar place names occur in several parts of Germany. This question explicitly connects the county name with Saarland. Use that state reference to orient yourself.
County overview · Landkreis Neunkirchen ↗Landkreis Neunkirchen → Saarland; county administration → Ottweiler.
Separate the territorial unit’s name from its administration’s address.
Three words for this question
- der Landkreiscounty
- A local territorial unit containing several towns and municipalities.
- der Verwaltungssitzadministrative seat
- The locality where a territorial unit’s administration is based.
- namensgebendgiving its name
- Neunkirchen gives the county its name.