Constipation: Given the patient's age, history of dementia, and symptoms of abdominal distention and dull, intermittent pain, constipation is a common and likely cause. The pain being elicited on palpation and with movement also supports this diagnosis, as constipation can cause discomfort that worsens with movement.

Diverticulitis: Although the pain is in the right lower quadrant (RLQ), diverticulitis typically presents with left lower quadrant pain, it's still a consideration, especially in the elderly. The presence of diarrhea and abdominal distention could be related to diverticulitis, especially if there's an associated partial bowel obstruction.
Urinary Tract Infection (UTI) or Pyelonephritis: Given the patient's history of BPH, a UTI could cause abdominal pain, especially if it has ascended to the kidneys (pyelonephritis). However, the absence of typical urinary symptoms (dysuria, frequency, urgency) makes this less likely.
Gastroenteritis: The history of diarrhea could suggest a gastrointestinal infection. However, the absence of other symptoms like vomiting or fever makes this less likely.

Appendicitis: Although less common in the elderly and the presentation might be atypical (e.g., less severe pain, fewer symptoms), appendicitis is a critical diagnosis not to miss due to its potential for severe complications if not promptly treated.
Bowel Obstruction: This could be a complication of constipation or another process like a tumor. The symptoms of abdominal distention and pain are consistent with this diagnosis, and it requires urgent attention.
Mesenteric Ischemia: This is a life-threatening condition that can present with abdominal pain out of proportion to the physical examination findings. The elderly are at higher risk, and the presence of atrial fibrillation (not mentioned but common in this age group) or other cardiac conditions increases the risk.

Inflammatory Bowel Disease (IBD): While IBD (Crohn's disease or ulcerative colitis) can present at any age, it is less common in the elderly without a prior history.
Intussusception: A condition where a part of the intestine telescopes into another, which can cause intermittent abdominal pain and is more common in children but can occur in adults, often with a lead point like a tumor.
Elderly-onset Crohn's disease or Ulcerative Colitis: Although rare, it's essential to consider in the differential diagnosis, especially if other common causes are ruled out.